Choose the wrong option

The correct answer (the wrong option among the choices) is:
"Gravitational mass of a particle like proton can depend on the presence of neighouring heavy objects but the inertial mass cannot"

Let us analyze each option step-by-step to understand why this statement is incorrect (and thus the correct choice for this question):

1. Understanding Inertial and Gravitational Mass:
Inertial mass is a measure of an object's resistance to acceleration when a force is applied (from Newton's second law, $F=m_{i}a$). Gravitational mass determines the strength of the gravitational force experienced by or exerted by the object in a gravitational field (from Newton's law of gravitation, $F=\frac{G{m}_{g}M}{r^2}$). Thus, the definition given in the first option is scientifically accurate.

2. Equivalence of Inertial and Gravitational Mass:
The equality of inertial mass ($m_i$) and gravitational mass ($m_g$) is an established experimental fact, verified to extremely high precision (e.g., via Eötvös-type experiments). This experimental equivalence forms the foundation of Einstein's Equivalence Principle in General Relativity.

3. Acceleration due to Gravity:
If we equate the gravitational force to the inertial force acting on a falling body:
$m_{i}a=\frac{GM{m}_g}{R^2}$
Solving for acceleration $a$ gives:
$a=\left(\frac{m_g}{m_i}\right)\frac{GM}{R^2}$
Because experimental evidence shows $m_g=m_i$, the ratio $\frac{m_g}{m_i}=1$, meaning the acceleration due to gravity $a=g=\frac{GM$ $}{R^2}$ is indeed the same for all bodies regardless of their mass.

4. Dependence on Neighboring Objects:
Both gravitational mass and inertial mass are intrinsic properties of a particle (such as a proton) and do not depend on the presence of neighboring heavy objects. Saying that the gravitational mass can depend on neighboring objects while the inertial mass cannot violates the equivalence principle and fundamental physical observations. Therefore, this statement is false, making it the correct option to choose.